Levy's phenomenon for entire functions of several variables

TL;DR

This paper extends Levy's phenomenon, originally observed for entire functions of one variable, to functions of several variables, showing the constant in Wiman's inequality can be improved similarly.

Contribution

It proves Levy's phenomenon applies to entire functions of multiple variables, answering a question posed by Goldberg and Sheremeta in 1996.

Findings

Levy's phenomenon holds for entire functions of several variables.

The constant 1/2 in Wiman's inequality can be replaced by 1/4.

The result confirms Levy's phenomenon is not limited to single-variable functions.

Abstract

For entire functions $f(z)=\sum_{n=0}^{+\infty}a_nz^n, z\in {\Bbb C},$ P. L${\rm \acute{e}}$vy (1929) established that in the classical Wiman's inequality $M_f(r)\leq\mu_f(r)\times $ $\times(\ln\mu_f(r))^{1/2+\varepsilon},\ \varepsilon>0,$ which holds outside a set of finite logarithmic measure, the constant 1/2 can be replaced almost surely in some sense, by 1/4; here $M_f(r)=\max\{|f(z)|\colon |z|=r\},\ \mu_f(r)=\max\{|a_n|r^n\colon n\geq0\},\ r>0. $ In this paper we prove that the phenomenon discovered by P. L${\rm\acute{e}}$vy holds also in the case of Wiman's inequality for entire functions of several variables, which gives an affirmative answer to the question of A. A. Goldberg and M. M. Sheremeta (1996) on the possibility of this phenomenon.